The present invention relates to interactions between marine vessels (e.g., steadily moving ships) and environmental waves, more particularly to methodologies for modeling such interactions.
The past few decades have seen great efforts toward the development of ship motion models. Generally, the ship motion models have been developed via algorithms or formulae based on either linear theory (e.g., Green function) or weakly nonlinear theory. Two major problems hinder the application of these conventional models, viz.: (i) their inability to model strongly nonlinear interactions (e.g., wherein the nonlinear interactions are comparable to or stronger than the linear variations) among surface waves, and among surface waves and ship bodies; (ii) the computationally expensive nature of the backbone algorithms, particularly in solving global wave-like solutions. Therefore, the conventional models are limited to the “mild-condition” applications; that is to say, the conventional models are significantly less than ideal (and may even be impossible) for solving extreme problems, such as storm surges, strong incident waves, and high speed ships. In addition, due to computational costs, the conventional models tend to be less likely than desired to meet real-time or near real-time application requirements.
Lord Kelvin first developed a mathematical description of wave pattern due to a point pressure distribution, the classical V-shaped wave pattern that bears his name; see W. Thompson, “On Ship Waves,” Proc. Inst. Mech. Eng., 1887, Reprint, 1891, in Popular Lectures and Addresses, 3:450-500, MacMillan, London; W. Thompson, “On the Waves Produced by a Single Impulse in Water of Any Depth, or in a Dispersive Medium,” Proceedings of the Royal Society of London Series A 42:80-85, 1887.
Michell developed a thin-ship theory for the wave resistance of a ship; see J. H. Michell, “The Wave Resistance of a Ship,” Phil. Mag. 45:106-23, 1898.
Havelock extended Michell's thin ship theory to predict the wave pattern about the ship; see T. H. Havelock, “The Propagation of Groups of Waves in Dispersive Media, with Application to Waves on Water Produced by a Traveling Disturbance,” Proceedings of the Royal Society of London Series A, 81:398-430, 1908; T. H. Havelock, “Wave Patterns and Wave Resistance,” Trans. Inst. Nay. Arch. 76:430-46 (1934).
Tuck developed slender body theory for a ship and the wave field near the ship. See E. O. Tuck, “A Systematic Asymptotic Expansion Procedure for Slender Ships,” Journal of Ship Research 8(1):15-23, 1964; E. O. Tuck, “On Line Distributions of Kelvin Sources, Journal of Ship Research 8(2):45-52, 1964.
Two families of linearized free surface solutions satisfying the exact body boundary condition have been developed, viz., the Neumann-Kelvin methods and Dawson the methods. The Neuman-Kelvin methods linearize the free-surface flow about the free-stream velocity. The Dawson methods linearize the problem about the double-body flow.
The Havelock singularity solution of the Neumann-Kelvin formulation of the ship-wave problem reduces the problem to the solution of an integral equation on the body surface by using a Green function that satisfies explicitly the Laplace equation and the linearized free-surface condition. Doctors together with Beck, and Scragg, have developed Havelock-Green function-based methods for computing the solution of the Kelvin-wake problem; see L. J. Doctors and R. F. Beck, “Numerical Aspects of the Neumann-Kelvin Problem,” Journal of Ship Research 31:1-13, 1987; L. J. Doctors and R. F. Beck, “Convergence Properties of the Neumann-Kelvin Problem for a Submerged Body,” Journal of Ship Research 31:227-34, 1987; A. M. Reed, J. G. Telste and C. A. Scragg, “Analysis of Transom Stern Flows,” Proceedings of the 18th Symposium on Naval Hydrodynamics, Ann Arbor, Mich., pp 207-19, Washington, D.C.: Natl. Acad. Press, 1990.
Doctors & Beck and Scragg solve the full Neumann-Kelvin problem for the strengths of the singularities. Telste (See A. M. Reed, J. G. Telste and C. A. Scragg, “Analysis Of Transom Stern Flows,” Proceedings of the 18th Symposium on Naval Hydrodynamics, Ann Arbor, Mich., pp 207-19, Washington, D.C.: Natl. Acad. Press, 1990; J. Telste and A. M. Reed, “Calculation of Transom Stern Flows,” Proceedings of the 6th International Conference on Numerical Ship Hydrodynamics, Iowa City, Iowa, pp 79-92, Washington, D.C.: Natl. Acad. Press, 1993) employs Rankine singularities to solve the Neumann-Kelvin problem, which necessitates distributing Rankine singularities on the free surface as well as on the hull surface.
Dawson methods (See C. W. Dawson, “A Practical Computer Method for Solving Ship Wave Problems,” Proceedings of the 2nd International Conference on Numerical Ship Hydrodynamics, Berkeley, Calif., pp 30-38, Berkeley, Calif.: Univ. Calif. Berkeley, 1977) solve for the Kelvin wake in a two-stage process. The first step involves the solution of a double-body problem where the ship's hull is reflected in the free surface and the flow is predicted as though the body and its reflected image are submerged infinitely deep in the fluid. The second stage of the solution involves solving a free-surface problem that is linearized around the double-body solution. This second solution involves singularities on the free surface as well as on the body. Compared to the Havelock Green function-based methods, the Dawson methods trade a much simpler Green function for a much larger computational domain with many more unknowns.
Cheng (1989), Kim et al. (1989), Sclavounos & Nakos (1988), and Nakos & Sclavounos (1990) have developed Dawson methods for solving the Kelvin-wake problem using Rankine sources. See B. H. Cheng, “Computations of 3D Transom Stern Flows,” Proceedings of the 5th International Conference on Numerical Ship Hydrodynamics, Washington, D.C., pp 581-92, Washington, D.C.: Natl. Acad. Press, 1989; Y-H Kim, S. H. Kim and T. Lucas, “Advanced Panel Method for Ship Wave Inviscid Flow Theory (SWIFT),” David Taylor Research Center (DTRC; now NSWCCD), Ship Hydromechanics Department, R & D Report DTRC-89/029, West Bethesda, Md., 66 pp, 1989; P. D. Sclavounos and D. E. Nakos, “Stability Analysis of Panel Methods for Free-Surface Flows with Forward Speed,” Proceedings of the 17th Symposium on Naval Hydrodynamics, Den Haag, Netherlands, 1988; D. Nakos and P. D. Sclavounos, “Ship Motions by a Three-Dimensional Rankine Panel Method,” Proceedings of the 18th Symposium on Naval Hydrodynamics, pp 21-41, Ann Arbor, Mich., 1990.
Scragg & Talcott (1990) and Scragg (1999) have developed a “Havelock-Dawson” approach that employs Rankine singularities on the hull and Havelock singularities on the free surface in a limited region of the undisturbed free surface near the body. See C. A. Scragg. and J. C. Talcott., “Numerical Solution of the “Dawson” Free-Surface Problem Using Havelock Singularities,” Proceedings of the 18th Symposium on Naval Hydrodynamics, Ann Arbor, Mich., pp 259-71, Washington, D.C.: National Academy Press, 1990; C. A. Scragg, “On the Use of Free-Surface Distributions of Havelock Singularities,” Proceedings of the 14th International Workshop on Water Waves and Floating Bodies, Port Huron, Mich., 4 pp, Ann Arbor, Mich.: Univ. Mich., Dep. Naval Arch. Mar. Eng., 1999. The Havelock-Dawson method allows one to panel only a limited region of the free surface because the singularities satisfy both the linearized free surface and the radiation boundary conditions, resulting in a “self-limiting” distribution of singularities on the free surface. The Havelock-Dawson approach is computationally very efficient.
Most recently, fully nonlinear free surface flow solutions have been developed based on both potential flow and on RANS formulations. Raven (1996, 1998), Subramani (2000), and Wyatt (2000) have published nonlinear potential flow methods for predicting the wave field around ships. See D. C. Wyatt, “Development and Assessment of a Nonlinear Wave Prediction Methodology for Surface Vessels,” Journal of Ship Research 44:96-107, 2000; A. K. Subramani, “Computations of Highly Nonlinear Free-Surface Flows, with Applications to Arbitrary and Complex Hull Forms,” PhD thesis, Dep. Naval Arch. Mar. Eng., Univ. Mich., Ann Arbor, Mich. 127 pp, 2000; H. C. Raven, A Solution Method for the Nonlinear Ship Wave Resistance Problem, Wageningen, Netherlands: Marin, 220 pp, 1996; H. C. Raven, “Inviscid Calculations of Ship Wave Making—Capabilities, Limitations, and Prospects,” Proceedings of the 22nd Symposium on Naval Hydrodynamics, Washington, D.C., pp 738-54, Washington, D.C.: Natl. Acad. Press, 1998.
All three authors in the preceding paragraph employ Rankine singularities and desingularize the free surface. Several authors have published the results from free-surface RANS codes that can solve the steady-ship wave problem. See A. Arabshahi, M. Beddhu, W. Briley, J. Chen, A. Gaither, et al., “A Perspective on Naval Hydrodynamic Flow Simulations,” Proceedings of the 22nd Symposium on Naval Hydrodynamics, Washington, D.C., pp 920-34, Washington, D.C.: Natl. Acad. Press, 1998; see also, R. Wilson, E. Paterson, and F. Stern, “Unsteady RANS CFD Method for Naval Combatants in Waves,” Proceedings of the 22nd Symposium on. Naval Hydrodynamics, Washington, D.C., pp 532-49, Washington, D.C.: Natl. Acad. Press, 1998.
These codes solve the field equations using a finite volume or finite difference scheme. Each satisfies the full nonlinear free-surface boundary condition by employing some type of upstream differencing scheme on the free surface. The fact that the fluid volume must be re-gridded to track the nonlinear free-surface deformation adds a significant complication to the iteration scheme and adds significantly to the computation time. Solutions using nonlinear free-surface RANS codes such as these can take 40-80 hours or more to compute on a state-of-the-art multiprocessor super computer.
In parallel with the above studies, effort has recently been focused on interactions between ship waves and ambient waves because they cannot be well described by linear superposition (See, e.g., M. St. Denis and W. J. Pierson, “On the Motions of Ships in Confused Seas,” SNAME Transactions, Vol. 61, 1953); Y. Liu, D. G. Dommermuth, and D. K. P. Yue, “A high-order spectral method for nonlinear wave-body interactions,” Journal of Fluid Mechanics, 245:115-136, 1992) first used a high-order spectral method to study nonlinear interactions among surface waves and ship. Their model was developed originally by Dommermuth and Yue (See D. G. Dommermuth and D. K. P. Yue, “Numerical Simulations Of Nonlinear Axisymmetric Flows With A Free Surface,” Journal of Fluid Mechanics 209, 57, 1987) to study nonlinear gravity wave interactions based on the third-order Zakharov equation. See R.-Q. Lin, and W. Perrie, “A New Coastal Wave Model, Part III: Nonlinear Wave-Wave Interaction for Wave Spectral Evolution,” Journal of Physical Oceanography, 27:1813-26, 1997; D. R. Crawford, B. M. Lake, P. G. Saffman, and H. C. Yuen, “Stability of Weakly Nonlinear Deep-Water Waves in Two and Three Dimensions,” Journal of Fluid Mechanics, 105:177-191, 1982.
However, the Zakharov Equation, and the Hasselmann Equation (See S. Hasselmann and K. Hasselmann, “A Symmetrical Method of Computing the Nonlinear Transfer in a Gravity-Wave Spectrum,” Hamb. Geophys. Einzelschrifien Reihe A Wiss. Abhand, Vol. 52, 138 pp, 1981), which is similar to the Zakharov equation, are both derived using perturbation methods, which implies that the equations are not appropriate for problems with large wave steepness. For example, the maximum wave steepness for the Zakharov equation is 0.3, with a 10 percent error level. With the similar error, the Hasselmann equation only allows wave steepness up to 0.06; see R.-Q. Lin and W. Perrie, “A New Coastal Wave Model, Part III: Nonlinear Wave-Wave Interaction for Wave Spectral Evolution,” Journal of Physical Oceanography, 27:1813-26, 1997.
In addition, the model of Liu et al. (1992) can only solve interactions between a cylinder and surface waves, while ships have much more complicated geometries. The solutions of Liu et al. can be extended to more complicated problems by applying conformal mappings, but this imposes constraints on the flow properties and on the ship geometries. In order to resolve arbitrary ship shapes, boundary element methods are introduced in modeling finite amplitude ship-wave interactions. See, e.g., W. M. Lin and D. K. P. Yue, “Numerical Solution for Large-Amplitude Ship Motions in Time-Domain,” Proceedings of the 18th Symposium on Naval Hydrodynamics, U. Michigan, Ann Arbor, Mich., 1990; W. M. Lin, M. J. Meinhold, N. Salvesen, and D. K. P. Yue, “Large-Amplitude Motions and Wave Load for Ship Design,” Proceedings of the 20th Symposium on Naval Hydrodynamics, U. California, Santa Barbara, Calif., 1994; Lin and Yue, 1990; Liu et al., 1992, 1994; M. Xue, “Three-Dimensional Fully-Nonlinear Simulations of Waves and Wave Body Interactions,” Ph.D. Thesis, Dept Ocean Engineering, MIT, 1997; M. Xue, H. Xü, Y. Liu, and D. K. P. Yue, “Computations of Full Nonlinear Three-Dimensional Wave-Wave and Wave-Body Interactions. Part I. Dynamics of Steep Three-Dimensional Waves,” Journal of Fluid Mechanics, 438:11-39. 2001; Xue, 1997; Xue et al, 2000; Y. Liu, M. Xue, and D. K. P, Yue, “Computations of Fully Nonlinear Three-Dimensional Wave-Wave Wave-Body Interactions, Part 2: Nonlinear Waves and Forces,” Journal of Fluid Mechanics, 438:41-66, 2000.
Though the boundary element method (and other local methods) is very good for resolving flow near the boundaries of arbitrary ships, it is not very efficient in resolving the waves away from the ship. This problem can be very serious when ambient waves are present. In this case, fine-scale grids are necessary to resolve small-scale waves and wave-wave interactions. Insufficient resolution can easily result in numerical instabilities. On the other hand, spectral methods can take advantage of the wave-like motions at the surface away from the ship. In particular, by selecting appropriate base wave functions for spectral expansion, one can easily model surface waves and wave-wave interactions with high computational efficiency. Furthermore, one can model wave-breaking mechanisms that are important in studying finite amplitude wave-wave interactions. Because of these numerical advantages, ship hydrodynamics models based on spectral methods have been developed. For example, Liu et al. (1992) developed a spectral model to study interaction between incident waves and a cylinder (an idealized ship).
However, the spectral method is not very efficient in solving for the local flow structures near the ship. For example, a very fine ship will generate very small-scale flow near the ship. Resolving this small-scale flow with spectral methods requires a very high truncation order, not necessary for the flow away from ship. In addition, traditional periodic boundary conditions used in modeling wave-wave interactions (See, e.g., R.-Q Lin and W. Kuang, “Nonlinear Wave-Wave Interactions of Finite Amplitude Gravity Wave,” Recent Developments in Physical Oceanography 8:109-116, 2004) are not applicable to ship-wave interaction problems. An example is the case of a ship moving in calm water where there are no waves ahead of the ship but waves extend behind the ship to infinity, where there can be no periodic far-field boundary conditions.
In ship-wave hydrodynamics, the flow is assumed to be an incompressible, inviscid potential flow. Therefore, singularities occur in solving flow at the ship boundary, particularly at bow and stern points. Resolving the flow near the ends of the ship is one of the important problems in ship hydrodynamics, and is particularly critical for correct evaluation of the pressure on a moving ship. In the past, researchers developed various approximations and techniques to avoid the singularities. As previously mentioned herein, Michell (1898) developed thin ship theory, which was further refined by Havelock (1908, 1934). Tuck (1964) developed slender theory for a ship and the wave field near the ship. Similar methods were developed in (linear and nonlinear) studies using the exact ship boundary condition, e.g. Doctors & Beck (1987), Scragg (Reed et al. 1990), Telste (Reed et al. 1990, Telste & Reed 1993), and Liu et al. (2000). These approaches to solving the velocity potential at the ship boundary depend on certain “cut-off” domains (e.g., truncation in wave number space of integrals for the wave components of the potential, or truncation of the Fourier series of the spectral representation). In addition, computations with arbitrary ship geometries can be very expensive, because high numerical resolution is required to avoid numerical errors and instabilities.